

A180186


Triangle read by rows: T(n,k) is the number of permutations of [n] starting with 1, having no 3sequences and having k successions (0<=k<=floor(n/2)); a succession of a permutation p is a position i such that p(i +1)  p(i) = 1.


2



1, 1, 0, 1, 1, 0, 2, 3, 0, 9, 8, 3, 44, 45, 12, 1, 265, 264, 90, 8, 1854, 1855, 660, 90, 2, 14833, 14832, 5565, 880, 45, 133496, 133497, 51912, 9275, 660, 9, 1334961, 1334960, 533988, 103824, 9275, 264, 14684570, 14684571, 6007320, 1245972, 129780, 5565
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OFFSET

0,7


COMMENTS

Row n has 1+floor(n/2) entries.
Sum of entries in row n is A165961(n).
T(n,0)=d(n1).
Sum(k*T(n,k), k>=0) = A180187(n).
Contribution from Emeric Deutsch, Sep 07 2010: (Start)
T(n,k) is also the number of permutations of [n1] with k fixed points, no two of them adjacent. Example: T(5,2)=3 because we have 1432, 1324, and 3214.
(End)


LINKS

Table of n, a(n) for n=0..47.


FORMULA

T(n,k) = binom(nk,k)*d(nk1), where d(j) = A000166(j) are the derangement numbers.


EXAMPLE

T(5,2)=3 because we have 12453, 12534, and 14523.
Triangle starts:
1;
1;
0,1;
1,0;
2,3,0;
9,8,3;
44,45,12,1;
265,264,90,8;


MAPLE

d[0] := 1: for n to 51 do d[n] := n*d[n1]+(1)^n end do: a := proc (n, k) if n = 0 and k = 0 then 1 elif k <= (1/2)*n then binomial(nk, k)*d[n1k] else 0 end if end proc: for n from 0 to 12 do seq(a(n, k), k = 0 .. (1/2)*n) end do; # yields sequence in triangular form


CROSSREFS

Cf. A000166, A165961, A180187
Sequence in context: A137914 A098989 A175315 * A256294 A279412 A012399
Adjacent sequences: A180183 A180184 A180185 * A180187 A180188 A180189


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Sep 06 2010


STATUS

approved



